What is the optimal time complexity of solving Friendly pair?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Friendly pair?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Friendly pair?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Friendly pair?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Friendly pair

Learn how to solve the 'Friendly pair' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function is_friendly_pair(a, b) that takes two positive integers a and b and returns True if they form a friendly pair, or False otherwise. Two numbers form a friendly pair if they have the same abundancy index. The abundancy index of a number n is defined as sigma(n) / n, where sigma(n) is the sum of all divisors of n (including n itself). Two numbers are friendly if sigma(a) / a == sigma(b) / b. To avoid floating-point issues, compare by cross-multiplying: sigma(a) * b == sigma(b) * a.

Constraints

•1 <= a, b <= 10^5

Examples

Example 1

Input

a = 6, b = 28

Output

True

Explanation

sigma(6) = 1+2+3+6 = 12. sigma(28) = 1+2+4+7+14+28 = 56. Cross check: 12 * 28 = 336, 56 * 6 = 336. They are equal, so they are a friendly pair.

Example 2

Input

a = 30, b = 140

Output

True

Explanation

sigma(30) = 72, sigma(140) = 336. Cross check: 72 * 140 = 10080, 336 * 30 = 10080. Equal, so friendly pair.

Example 3

Input

a = 5, b = 10

Output

False

Explanation

sigma(5) = 6, sigma(10) = 18. Cross check: 6 * 10 = 60, 18 * 5 = 90. Not equal, so not a friendly pair.

Need a Hint?

Use simple arithmetic operators (like modulo `%`, division `//`), conditional checks, or loops to inspect number properties.

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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